Piet Mondrian was the very picture of modernity. Photographs of the boogie-woogie-loving neo-plasticist often show him bow-tied, angular and suave amid the rectangles that climbed the walls of his studio. The divisions and flat colours that filled his paintings seemed to migrate to every surface and multiply, rectangle upon rectangle, line upon line, proliferating in his studio mirrors and reflecting in the windows. Even the floor was divided by an arrangement of plain-coloured rugs. Among it all were his paintings, displayed on easels and walls, their own squares, rectangles and lozenges colliding with the shapes around them. Some were hung above head-height, up towards the ceiling.

None of the reconstructed studio's contents are actually Mondrian's: neither the paintings nor the coal scuttle, the pipe in its ashtray, the clean, square-edged brushes and the opera glasses. What did he need these little binoculars for in his cramped quarters? Maybe he looked through the wrong end, to see his paintings as if from afar. This is not so farfetched as it might seem. The reconstruction, first made for a 1995 exhibition at the Pompidou, gives us a vivid idea of Mondrian's ideas about space and form.

It is a surprise, perhaps, that Mondrian – like Barnett Newman – saw the studio as sanctuary as well as workshop. There he entertained other artists and potential collectors, and had himself photographed for newspapers and magazines. Sometimes, we glimpse him in a housecoat, pretending to paint for the camera. His studio became a kind of theatre, a stage set where he could act out being Mondrian, for himself and for his public.

Mondrian's Composition in Colour B, 1917. Photograph: Tate Liverpool

It was all of a piece with his paintings. How Mondrian organised his studio and adjacent living spaces demonstrated the way he wanted his art to go beyond painting and into the modern word. He related his work to more than architecture, the horizontals and the verticals of the city's grid, seeing it as universal. "And man?" he said. "He must be nothing in himself, just part of the whole. He thus no longer feels his individuality, he will be happy in the earthly paradise he has created."

Tate Liverpool's Mondrian and His Studios follows the artist from studio to studio during the mature phase of his career, from 1914 until his death in 1944. There is a touching moment when we come across the passenger list of the ship that in 1940 carried him to New York. Pier Head, where he stepped on board, is visible through a nearby window. It echoes the fact Mondrian's art, for all its flatness, is filled with distances, proximities and reversals. One of his greatest series, Pier and Ocean, has always seemed to be about exactly this.

Mondrian in his studio in Paris, 1933. Photograph: Tate Liverpool

Mondrian and Colour, a second exhibition at Turner Contemporary in Margate, traces the longer arc of his development: from its beginnings at the exhausted end of the Dutch landscape tradition through to the austere, beautiful and pared-down Lozenge Composition With Four Yellow Lines from 1933 – by way of his belated impressionist paintings, his symbolist phase and his flirtation with theosophy. While many visitors at Margate lose themselves among the moonlit windmills and backlit oaks and elms of his earlier work, I am transfixed by the deceptive simplicity of the Lozenge Composition. It still feels timeless in a way his landscapes don't. Both contained and expansive, it hangs high on the wall as Mondrian intended. There's nothing to it but yellow and white, four yellow bars of slightly uneven width and optical weight that disturb what otherwise would be an absolute symmetry.

It is in his later works that the hand-craftedness of Mondrian's paintings really counts. There's no avoiding the traces of his touch. The Dutchman was never purely mechanical. Though the exhibition is called Mondrian and Colour, it never goes into its actual complexities. Neither his whites nor his blacks are all the same. Did he mix his primaries or were they direct from the tube? Mondrian was as acutely aware of the effects of sheen, from matt to gloss, and the thickness and weight of the paint, as he was of colour and tonal values.

Up close, his art isn't nearly as clear and clean as it looks in reproduction. The pure white paint has cracked. The brush makes its slow negotiations, edge against edge, plane against plane. One unfinished painting in Liverpool has rough charcoal lines drawn among the painted bars, showing us how intuitive his working procedure really was. He found unity and division as much by sight and feel as by ruler and premeditation. All of this somehow humanises Mondrian's art, and more importantly slows us down when we look.

Taken together, the two exhibitions make for a patchy and incomplete retrospective of an artist whose work aspired to be universal. Mondrian has now become a brand as much as an aesthetic. Having a chain of American hotels named after him and his rectangles and lines decorating Yves Saint Laurent dresses and Kleenex boxes probably isn't quite what he had in mind. I prefer to imagine Mondrian dancing among his squares and rectangles in his studio, lost in the music and the world of his own creation.